| L(s) = 1 | − 2-s − 2.52·3-s + 4-s − 5-s + 2.52·6-s + 2.66·7-s − 8-s + 3.39·9-s + 10-s + 0.213·11-s − 2.52·12-s + 13-s − 2.66·14-s + 2.52·15-s + 16-s − 0.744·17-s − 3.39·18-s − 2.06·19-s − 20-s − 6.72·21-s − 0.213·22-s + 1.27·23-s + 2.52·24-s + 25-s − 26-s − 0.998·27-s + 2.66·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.45·3-s + 0.5·4-s − 0.447·5-s + 1.03·6-s + 1.00·7-s − 0.353·8-s + 1.13·9-s + 0.316·10-s + 0.0644·11-s − 0.729·12-s + 0.277·13-s − 0.711·14-s + 0.652·15-s + 0.250·16-s − 0.180·17-s − 0.800·18-s − 0.474·19-s − 0.223·20-s − 1.46·21-s − 0.0455·22-s + 0.265·23-s + 0.516·24-s + 0.200·25-s − 0.196·26-s − 0.192·27-s + 0.502·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 11 | \( 1 - 0.213T + 11T^{2} \) |
| 17 | \( 1 + 0.744T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 37 | \( 1 - 6.35T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 + 0.538T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 8.67T + 67T^{2} \) |
| 71 | \( 1 + 7.36T + 71T^{2} \) |
| 73 | \( 1 - 4.07T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.043899157199430368898659704179, −7.37873429720192597958376327694, −6.57539185555870517226446358947, −5.97875137254863726528397833240, −5.10249834472493694457867806955, −4.55021958983803807060937201893, −3.48917925516279355206434046052, −2.06044411436935169570414979318, −1.10903093294458217292974087119, 0,
1.10903093294458217292974087119, 2.06044411436935169570414979318, 3.48917925516279355206434046052, 4.55021958983803807060937201893, 5.10249834472493694457867806955, 5.97875137254863726528397833240, 6.57539185555870517226446358947, 7.37873429720192597958376327694, 8.043899157199430368898659704179
